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Interior angle sums of geodesic triangles and translation-like isoptic surfaces in Sol geometry

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arxiv 2405.05266 v1 pith:QUKCMF37 submitted 2024-04-24 math.MG

classification math.MG
keywords anglegeodesicsumstranslation-liketrianglesgeometriesinteriorisoptic
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abstract

After having investigated the geodesic triangles and their angle sums in Nil and $Sl\times\mathbb{R}$ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $\pi$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Moln\'ar in \cite{M97}.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Menelaus' and Ceva's theorems for translation triangles in Thurston geometries

    math.GT 2025-06 reject novelty 5.0 of 10

    Menelaus' and Ceva's theorems are formulated and proved for translation triangles in Nil, Sol, and ~SL2R spaces using geometry-specific simple ratios.

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